The first objective of the study is to develop statistical procedures that aid the planning and analysis of incomplete survival time. The second is to develop a unified procedure of diagnostics, the jackknife and the bootstrap for the Cox model and partial likelihood estimators. The study contains five aims. The first four topics consider statistical inference about different kinds of incomplete survival data. These are the mixture of survival time and other variables (aim 1), doubly-censored data (aim 2), truncated data with censoring (aim 3) and bivariate censored data (aim 4). Aims 1-4 are not only important in their own right, they can be applied to the AIDS data. Statistical inference for the long-term survivors and Huntington's disease will be developed. In many cases, in the experiments, a substantial proportion of the animals at some intoxicant levels do not die by the end of the experiment. The same situation will happen in AIDS incidence. A portion of infected individuals will never develop AIDS (or will develop AIDS very late). We model these data by involving the mixture of two populations. For an animal with covariate, we will use the logistic model for the proportion of long-term survivors and Cox model for the failure time of short-term survivors. A semi-nonparametric technique will be employed to propose statistical analyses. Linear rank statistic for doubly-censored data will be developed. These statistics can be applied to tumorigenicity experiments with animals. It is anticipated that the procedure will be statistically and computationally more efficient than those in current use. The extension of these statistics to the interval censored data will be investigated as well. The interval censored data will fit the Hershey hemophiliacs cohort with AIDS study naturally. Because the data of seroconversion could be only determined to have occurred in the interval between the last seronegative and the first seropositive serum sample. Truncation arises naturally when individuals come under observation only some known time after the time origin. For example, in the study of pediatric AIDS children who contract AIDS can only be identified if they develop AIDS before current calendar time. The major areas of research proposed for these kind of data are (i) testing the independency assumption between truncation of time and failure time, and (ii) providing statistical inference based on linear regression model as well as the Cox model. The properties of generalized self-consistent estimator of the survival function for bivariate censored data will be studied. It is anticipated that the estimator will be a genuine survival function and more efficient than those in current use. Bootstrap method is a well-recognized statistical tool even among the non-statisticians, but the bootstrap method is not very successful for the Cox model. In the last aim, we propose a unified and promising procedure of the resampling scheme for the Cox model and the maximum partial likelihood estimators. The same idea can also be used for diagnostic and jackknife. The most important feature is the computationally efficient, because the procedure only form the risk set once.